Dear Arminius, I'm certainly not going to answer your questions "why doesn't he say...?": Qing is a frequent and friendly contributor to MO and he will answer himself if he wants to.

Here is what I think is the consensus about your questions.

1. For a scheme regular definitely implies locally noetherian: De Jong 19.8.2

2. Birational necessitates neither noetherian nor reducedness conditions on schemes nor finite type assumptions on morphisms: De Jong 20.7.1

3. Qing's definition now makes perfectly good sense in view of 1) and 2). Desingularization is automatically of finite type because a proper morphism is of finite type by definition : De Jong 20.36.1

Bibliographical note I didn't want to give a long list of references for the definitions you ask about. I have only quoted De Jong and collaborators' monumental Stacks Project which is the most up-to-date reference and which is incredibly well thought-out.

Dear Arminius, I'm certainly not going to answer your questions "why doesn't he say...?": Qing is a frequent and friendly contributor to MO and he will answer himself if he wants to.

Here is what I think is the consensus about your questions.

1. For a scheme regular definitely implies locally noetherian: De Jong 19.8.2

2. Birational necessitates neither noetherian nor reducedness conditions on schemes nor finite type assumptions on morphisms: De Jong 20.7.1

3. Qing's definition now makes perfectly good sense in view of 1) and 2). Desingularization is automatically of finite type because a proper morphism is of finite type by definition : De Jong 20.36.1

Bibliographical note I didn't want to give a long list of references for the definitions you ask about. I have only quoted De Jong and collaborators' monumental [Stacks Project][1] which is the most up-to-date reference and which is incredibly well thought-out. Also De Jong is arguably the mathematician who has made the greatest progress on the resolution of singularities for schemes since Hironaka in 1964 . [1]: http://math.columbia.edu/algebraic_geometry/stacks-git/

Dear Arminius, I'm certainly not going to answer your questions "why doesn't he say...?": Qing is a frequent and friendly contributor to MO and he will answer himself if he wants to.

Here is what I think is the consensus about your questions.

1. For a scheme regular definitely implies locally noetherian: De Jong 19.8.2

2. Birational necessitates neither noetherian nor reducedness conditions on schemes nor finite type assumptions on morphisms: De Jong 20.7.1

3. Qing's definition now makes perfectly good sense in view of 1) and 2). I don't know if desingularization is automatically of finite type.

Bibliographical note I didn't want to give a long list of references for the definitions you ask about. I have only quoted De Jong and collaborators' monumental Stacks Project which is the most up-to-date reference and which is incredibly well thought-out.

Dear Arminius, I'm certainly not going to answer your questions "why doesn't he say...?": Qing is a frequent and friendly contributor to MO and he will answer himself if he wants to.

Here is what I think is the consensus about your questions.

1. For a scheme regular definitely implies locally noetherian: De Jong 19.8.2

2. Birational necessitates neither noetherian nor reducedness conditions on schemes nor finite type assumptions on morphisms: De Jong 20.7.1

3. Qing's definition now makes perfectly good sense in view of 1) and 2). Desingularization is automatically of finite type because a proper morphism is of finite type by definition : De Jong 20.36.1

Bibliographical note I didn't want to give a long list of references for the definitions you ask about. I have only quoted De Jong and collaborators' monumental Stacks Project which is the most up-to-date reference and which is incredibly well thought-out.